On symplectic automorphisms of elliptic surfaces acting on $mathrm{CH}_0$

Let $S$ be a complex smooth projective surface of Kodaira dimension one. We show that the group $mathrm{Aut}_s(S)$ of symplectic automorphisms acts trivially on the Albanese kernel $mathrm{CH}_0(S)_{mathrm{alb}}$ of the $0$-th Chow group $mathrm{CH}_0(S)$, unless the geometric genus and the irregularity satisfy $p_g(S)=q(S)in{1,2}$. In the exceptional case, the image of the homomorphism $mathrm{Aut}_s(S)rightarrow mathrm{Aut}(mathrm{CH}_0(S)_{mathrm{alb}})$ is either trivial or possibly isomorphic to $mathbb{Z}/3mathbb{Z}$. Our main arguments actually take care of the group $mathrm{Aut}_f(S)$ of fibration preserving automorphisms of elliptic surfaces $fcolon Srightarrow B$. We prove that, if $sigmainmathrm{Aut}_f(S)$ induces the trivial action on $H^{i,0}(S)$ for $i>0$, then it induces trivial action on $mathrm{CH}_0(S)_mathrm{alb}$. Also, if $S$ is additionally a K3 surface, then $mathrm{Aut}_f(S)cap mathrm{Aut}_s(S)$ acts trivially on $mathrm{CH}_0(S)_{mathrm{alb}}$.